Problem 34 - Chemistry The Central Science

Question

(a) In terms of the Bohr theory of the hydrogen atom, what process is occurring when excited hydrogen atoms emit radiant energy of certain wavelengths and only those wavelengths? (b) Does a hydrogen atom "expand" or "contract" as it moves from its ground state to an excited state?

Step-by-Step Solution

Answer

(a) When excited hydrogen atoms emit radiant energy of certain wavelengths and only those wavelengths, it represents the process of electron transitions between various energy levels in the hydrogen atom according to the Bohr theory. (b) A hydrogen atom "expands" as it moves from its ground state to an excited state, because the electron transitions to an orbit with a larger size, spending more time farther away from the nucleus.

Step 1: (Step 1: Understand energy levels in Bohr theory)

(According to Bohr's theory, electrons in a hydrogen atom move in orbits around the nucleus with certain allowed energy levels. When an electron transitions from a higher energy level to a lower energy level, the atom emits light with a specific wavelength. The emitted light corresponds to the energy difference between the two levels.)

Step 2: (Step 2: Relate emitted light to energy transitions)

(The energy difference between the two levels determines the wavelength of the emitted light, according to the following formula: \[\Delta E = E_{final} - E_{initial} = -13.6 \frac{eV}{n^2}\] where \(n\) is the principal quantum number of the energy level, and \(\Delta E\) is the energy difference. The energy difference is also related to the wavelength and frequency of emitted light by the equation: \[\Delta E = h \nu\] where \(h\) is Planck's constant and \(\nu\) is the frequency. Combining the two equations allows us to calculate the wavelength of the emitted light: \[\lambda = \frac{c}{\nu} = \frac{hc}{\Delta E}\] where \(\lambda\) is the wavelength and \(c\) is the speed of light.)

Step 3: (Step 3: Determine change in hydrogen atom size)

(As a hydrogen atom moves from the ground state (n=1) to an excited state (n>1), the electron spends more time farther away from the nucleus because the higher energy orbit is larger in size. This means that the hydrogen atom "expands" as it moves from the ground state to an excited state.)

Step 4: (Answer)

(a) When excited hydrogen atoms emit radiant energy of certain wavelengths and only those wavelengths, it represents the process of electron transitions between various energy levels in the hydrogen atom according to the Bohr theory. (b) A hydrogen atom "expands" as it moves from its ground state to an excited state, because the electron transitions to an orbit with a larger size, spending more time farther away from the nucleus.)

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe, consisting of only one proton and one electron. This single electron orbits the nucleus, much like the planets orbit around the sun. Bohr's model of the hydrogen atom is crucial for explaining how electrons behave in atoms. This model introduces the idea that electrons travel in specific paths, or orbits, and each orbit corresponds to a distinct energy level.

• The nucleus is at the center, with a positive charge due to the proton.
• The electron moves around the nucleus in distinct circular orbits.
• Each orbit is associated with a specific energy, which means electrons can only occupy certain energy levels.

In its ground state, the electron is in the closest possible orbit to the nucleus. When the atom absorbs energy, the electron jumps to a higher orbit, a process known as excitation. Conversely, when an electron falls back to a lower orbit, it releases energy in the form of light, which results in the emission of radiant energy. This fundamental interaction of electrons within the hydrogen atom forms the basis of our understanding of atomic structure.

Energy Levels

Energy levels are like rungs on a ladder that an electron can jump between. Each level is associated with a specific amount of energy. In Bohr's theory, these levels are quantized, meaning electrons can only exist in certain levels. This quantization is what allows for the emission of specific wavelengths of light.

• The lowest energy state an electron can occupy is called the ground state.
• Higher energy states, where electrons have absorbed energy and moved further from the nucleus, are called excited states.
• The principal quantum number, denoted as n, determines the energy level.

When transitioning between these levels, an electron either absorbs or emits energy. The formula \[ \Delta E = -13.6 \frac{eV}{n^2} \] determines the energy difference between levels, where is the principal quantum number. This energy difference corresponds to the frequency and wavelength of the light emitted or absorbed, characterizing the spectral lines unique to hydrogen.

Electron Transitions

Electron transitions are at the heart of how light is emitted in the hydrogen atom. When an electron changes its orbit, it moves between different energy levels. This movement is called a transition and can occur in two ways: emission and absorption.

• Emission: Occurs when an electron falls from a higher energy level to a lower one, releasing a photon of light.
• Absorption: Happens when an electron absorbs a photon of light and jumps from a lower energy level to a higher one.

The energy difference between the initial and final levels determines the wavelength of the emitted or absorbed light. The relationship between energy, frequency ( u ), and wavelength ( \lambda ) is expressed by the equation: off \[ \Delta E = h u = \frac{hc}{\lambda} \] where \(h\) is Planck's constant and \(c\) is the speed of light. This formula helps us understand why only certain wavelengths of light are emitted by a hydrogen atom: each transition corresponds to a specific energy change, producing unique spectral lines.